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arXiv:cond-mat/0011141v2 [cond-mat.dis-nn] 30 Jan 2001
Reducing nonideal to ideal coupling in random matrix description of
chaotic scattering: Application to the time-delay problem
Dmitry V. Savin1,2, Yan V. Fyodorov3, and Hans-J¨ urgen Sommers1
1Fachbereich Physik, Universit¨ at-GH Essen, 45117 Essen, Germany
2Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
3Department of Mathematical Sciences, Brunel University, Uxbridge, UB8 3PH, United Kingdom
(Received 8 November 2000; published in Phys. Rev. E 63 (March 2001), Rapid Communication)
We write explicitly a transformation of the scattering
phases reducing the problem of quantum chaotic scattering
for systems with M statistically equivalent channels at non-
ideal coupling to that for ideal coupling. Unfolding the phases
by their local density leads to universality of their local fluctu-
ations for large M. A relation between the partial time delays
and diagonal matrix elements of the Wigner-Smith matrix is
revealed for ideal coupling. This helped us in deriving the
joint probability distribution of partial time delays and the
distribution of the Wigner time delay.
PACS numbers: 05.45.-a, 24.60.-k, 73.23.-b
The random matrix theory (RMT) is generally ac-
cepted to be an adequate tool for describing various
universal statistical properties of quantum systems with
chaotic intrinsic dynamics, see Ref. [1] and references
therein. In particular, one can distinguish two variants
of the RMT approach allowing one to address the chaotic
nature of quantum scattering. The first one [2] considers
the scattering matrix S as the prime object without any
reference to the system Hamiltonian. The probability
distribution P(S) of S at the fixed energy E of incident
particles is chosen to satisfy a maximum entropy principle
and natural constraints which follow from the unitarity
and causality of S, and the presence (or absence) of the
time-reversal(TRS) and spin-rotation (SRS) symmetries,
P(S) ∝
????
det(1 −¯S†¯S)
det(1 −¯S†S)2
????
(βM+2−β)/2
.(1)
Such a distribution is known as the Poisson kernel [3]
and uses the phenomenological average (or optical) S-
matrix S(E) as the set of input parameters. Without
loss of generality¯S can be considered as diagonal [4].
P(S) depends also on the number of scattering channels
M and the symmetry index β [β=2 for a system with
broken TRS, and β=1(4) if the TRS is preserved and the
SRS is present (absent)].
The approach proved to be a success for extracting
many characteristics important in the theory of meso-
scopic transport [5]. However, correlation properties of
the S-matrix at close values of energy E as well as spec-
tral characteristics of an open system related to the so-
called resonances turn out to be inaccessible in the frame-
work of such an approach, essentially because of the one-
energy nature of the latter. To address such quantities
one needs to consider the HamiltonianˆH of the quantum
chaotic system as the prime building block of the the-
ory. It amounts to treatingˆH as a large N × N random
matrix of appropriate symmetry and relating S to the
Hamiltonian by means of standard tools of the scatter-
ing theory [6,7]. This idea supplemented with the super-
symmetry technique of ensemble averaging [8] resulted
in advance in calculating S-matrix correlation functions
[6,9] and many other related characteristics such as, e.g.,
time delays [10–12], see Refs. [11,1] for a review.
In the limit N→∞ one can prove [13] the equivalence
of both mentioned approaches by deriving the Poisson
kernel (1) from the Hamiltonian approach (see also Ref.
[11]), with the average S-matrix being
S(E)ab=1 − γa[iE/2 + πν(E)]
1 + γa[iE/2 + πν(E)]δab,(2)
independent of β. Here, the average density of states
ν(E)=π−1?1−(E/2)2determines the mean level spac-
ing ∆=(νN)−1of the closed system, and phenomenolog-
ical constants γc>0 characterize the coupling strength to
continuum in different scattering channels (c=1,...,M).
The particular case of ideal coupling,¯S=0 [when the
transmission coefficients equal unity for all channels, see
Eq. (6) below], plays an especiallly important role for the
S-matrix approach [14]. Equation (1) simplifies then to
P0(S)=const, which is invariant under the transforma-
tions of S, leaving the measure invariant. Such a situ-
ation corresponds to the so-called Dyson’s circular en-
semble (CE) of unitary matrices and is much simpler to
handle analytically.
A general situation of nonideal coupling,¯S ?=0, turns
out to be much more complicated. It is natural to ex-
pect, however, that results obtained for the case of non-
ideal coupling could be related to those at ideal coupling.
Although many useful ideas around such a relation were
discussed in the literature [2,13,11,15], we are not aware
of explicit relations, to the best of our knowledge.
In this Rapid Communication we consider the most
simple but physically important case of statistically
equivalent channels.We demonstrate the validity of
the following simple statement (and discuss several ap-
plications of it): Let S(E) = Uˆ s(E)U†, where ˆ s(E) =
diag(e2iδ1(E),...,e2iδM(E)), be the random S-matrix at
the energy E, the distribution of which is given by the
1
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Poisson kernel (1) with an explicit parameterization of
S(E) from Eq. (2) (γc=γ for all c). Then for every E the
transformation of the eigenphases δc(E)
?
πν(E)
φc= arctan
1
?1
γtanδc(E) +E
2
??
(3)
maps them to the eigenphases φcof the random scatter-
ing matrix, the distribution of which is given by the CE of
the same symmetry. In particular, the joint probability
density function (JPDF) of φcis [1]
?
a<b
p0
?{φc}?∝
??e2iφa− e2iφb??β.(4)
The matrix U of energy dependent eigenvectors uni-
formly distributed in the orthogonal, unitary, or symplec-
tic group (for β=1,2, or 4, respectively) is not affected by
map (3). This is a consequence of statistical equivalence
of the scattering channels.
The suggested transformation was first noticed and
verified in Ref. [11] for the case of broken TRS (β=2)
and further exploited in Ref. [12]. It can be easily gener-
alized to the other symmetry classes as follows. We cal-
culate first the Jacobian of the transformation (3). After
a simple algebra it can be represented as
∂φc
∂δc
=
cos2φc
πνγ cos2δc
=
T
∗e2iδc|2.
|1 − S
(5)
where
T(E) = 1 − |S(E)|2=
4γ πν(E)
1 + γ2+ 2γ πν(E). (6)
is the energy dependent transmission coefficient [6]. We
note that exactly the same factors as Eq. (5) appear in
Eq. (1). Employing the identity
??e2iφa− e2iφb??=
we substitute Jacobians (5) into Eq. (4) and, making use
of the identity?M
p?{δc}?∝
a<b
??e2iδa− e2iδb??
????
cosφacosφb
πνγ cosδacosδb
????, (7)
a<bfafb= (?
??e2iδa− e2iδb??β
cfc)M−1, arrive at
?
M
?
c=1
????
∂φc
∂δc
????
β(M−1)/2+1
. (8)
With Eq. (5) taken into account, we immediately rec-
ognize in this expression the JPDF of the eigenphases
corresponding to Poisson kernel (1). Due to the scalar
nature of transformation (3) it does not change the ma-
trix U of eigenvectors.
Let us start with considering the mean density ρ(δ) of
scattering (eigen)phases at arbitrary coupling. It is self-
evident that the phases in the CE (i.e. for the case S=0)
are uniformly distributed on the unit circle, the average
density being merely ρ0(φ) =(1/M)?
cδ(φ−φc)=1/π.
The corresponding density for S?=0 is not constant. In-
deed, using the identity ρ(δ)dδ=ρ0(φ)dφ, we see that
????
Although simple, this relation is an important one and es-
tablishes the physical meaning of the Jacobians of trans-
formation (3) relating them to the corresponding densi-
ties of the scattering phases. Density (9), being expressed
in terms of S only, does not depend on the particular
choice of S used in the derivation as long as the average
S-matrix is proportional to the unit matrix.
It is instructive to look at Eq. (8) in the limit of large
number of channels when the typical difference δa−δb∼
1/M ≪ 1.Then one can expand δc= δ0+˜δc (˜δc≪1)
around, say, δ0. The leading contribution is given by
p?{˜δc}?∝?
further goes to p0
distribution (4) of the CE upon the proper rescaling of
the phases,
ρ(δ) =1
π
∂φ
∂δ
????=
T
|1 − S
∗e2iδ|2.(9)
a<b|˜δa−˜δb|β?
c|∂φc/∂δc|β(M−1)/2+1
δ0
?{˜φc}?∝?
, which
a<b|˜φa−˜φb|βand agrees with
˜φc= |∂φc/∂δc|δ0˜δc= πρ(δ0)˜δc. (10)
We see that in the limit M ≫ 1 the local fluctuations
of the phases unfolded by their local density turn out to
be uniformly described by the CE at arbitrary coupling
strength. Such a universality in statistics of phases of
random unitary (scattering) matrices has much in com-
mon with that typical for eigenvalues of random Hamil-
tonian matrices [1] and is in agreement with results of
realistic numerical simulations for M = 23 [16].
Let us now consider an application of the same ideas to
the time-delay problem, where such a universality reveals
itself explicitly. Following the original wave-packet anal-
ysis by Eisenbud, Wigner and Smith [17] it is natural to
define [11] the partial time delays via the energy deriva-
tive of the scattering phases, τc= 2¯ h∂δc/∂E. Their sta-
tistical properties have been studied in much detail in
the framework of the Hamiltonian approach for the case
of broken [11] and preserved TRS as well as in the whole
crossover region of gradually broken TRS [12]. Recently,
some of these predictions were successfully verified on the
model of a quantum Bloch particle chaotically moving in
a superposition of ac and dc fields [18].
In particular, the mean density of partial time delays
P(τ)=(1/M)?
ple at ideal coupling, T=1, when it reads as
cδ(τ−τc) turns out to be especially sim-
P0(t=τ/tH) =(β/2)βM/2
Γ(βM/2)
e−β/2t
tβM/2+2,(11)
with tH=2π¯ h/∆ being the Heisenberg time.
Eq. (10), the partial time delays at ideal and nonideal
coupling (τ(0)
c
and τc, respectively) are simply related as
Due to
τ(0)
c
= 2¯ h∂φc/∂E = πρ(δc)τc.(12)
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Here, we have neglected the smooth nonresonant depen-
dence of ρ(δ) on E. Since the phase and its derivative
(the partial time delay) are uncorrelated quantities in the
CE [19], their joint distribution factorizes: ? p0(φ,τ(0))=
S ?=0. The relation ? p0(φ,τ(0))dφdτ(0)= ? p(δ,τ)dδ dτ be-
of partial time delays at nonideal coupling as
?π
0
∂(δ,τ)
?π
0
(1/π)P0(τ(0)).This is not the case for ? p(δ,τ), when
tween them allows us, however, to represent the density
P(τ) =
dδ
????
∂(φ,τ(0))
????? p0
?φ(δ),τ(0)(δ,τ)?
=
dδ
π[πρ(δ)]2P0(πρ(δ)τ).(13)
One can easily convince oneself [20] that such a formula
reproduces in every detail the expression obtained in Ref.
[11] by means of supersymmetry calculations. It is worth
mentioning that the density of phases (9) is independent
of the underlying symmetry and therefore Eq. (13) is also
valid for the crossoverregime of partly broken TRS. [Note
that in the crossover regime P0(t) is a slightly more com-
plicated function, see Ref. [12]].
Expression (13) is the proper one for generalization to
the JPDF of the partial time delays, w?{τc}?. Before
doing this, we first establish a useful relation between
τc and the matrix elements of the Wigner-Smith time-
delay matrix Q = −i¯ h(∂S/∂E)S†[17]. Writing S in the
eigenbasis representation as S = Uˆ sU†, one obtains
U†QU = −i¯ h∂ˆ s
∂Eˆ s†+ i¯ h
?
ˆ s,U†∂U
∂E
?
ˆ s†, (14)
where [,] denotes the commutator. The matrix ˆ s being
diagonal, the diagonal elements of the second term in
Eq. (14) are zero, whereas the first term is exactly the
diagonal matrix of the partial time delays. Thus, the
partial time delays coincide with the diagonal elements
of the time-delay matrix taken in the eigenbasis of the
scattering matrix,
τc= [U†QU]cc. (15)
The physical meaning of the diagonal elements of the
time-delay matrix is well known: they describe the time
delay of a wave-packet incident in a given channel [17,21].
Thus, relation (14) sheds more light on the physical
meaning of the somewhat formally defined partial time
delays. In particular, one expects that for the case of
ideal coupling the inherent rotational invariance of the
problem makes all the basises statistically equivalent and
thus the JPDF of diagonal elements of the Q-matrix
should coincide with that of partial time delays.
The latter claim can be substantiated as follows. Fol-
lowing the insightful paper [19], it is convenient to con-
sider the “symmetrized” time-delay matrix Qs
Qs= S−1/2QS1/2= −i¯ hS−1/2∂S
∂ES−1/2.(16)
This similarity transformation unveils the symmetry
which is hidden in Q: Qs is already a real symmetric
(hermitian, or quaternion self-dual) matrix for β = 1, (2,
or 4). In the eigenbasis of S the diagonal elements of Qs
and those of Q coincide. Moreover, in the case of chaotic
scattering with ideal coupling, the matrix Qsturns out to
be statistically independent of S, their joint probability
density being?P0(S,Qs)=P0(S)W0(Qs), where
W0(Qs) ∝ θ(Qs)det(Qs)−3βM/2−2+βe−(β/2)tHtrQ−1
s
(17)
is the probability density of the time-delay matrix [19].
The latter is manifestly invariant under the choice of the
basis for Qsproving the above statement on the relation
between statistics of partial time delays and diagonal el-
ements of the Wigner-Smith matrix.
To find the corresponding JPDF w0
integrate out all off-diagonal elements of Qs which is a
hard problem in general. For the case of unitary sym-
metry, β = 2, one can perform the job by splitting the
integration into that over the matrix ˆ q = diag(q1,...,qM)
of eigenvalues of Qsand that of the eigenvectors, V ,
?
with ∆(ˆ q) =?
stands for the remaining integral over the unitary group
which can be done, following Ref. [22], by means of the
famous Itzykson-Zuber formula [23]. Finally, we find it
more convenient to define the generating function of par-
tial time delays rather than the JPDF itself and obtain
?{τc}?one has to
wu
0
?{τc}?∝d[ˆ q]θ(ˆ q)∆2(ˆ q)
det(ˆ q)3Me−tHtrˆ q−1Q?{τc}?,(18)
a<b(qa− qb) being the Vandermonde de-
terminant. Here Q?{τc}?=?d[V ]?M
c=1δ?τc−(V ˆ qV†)cc
?
?
e−i(k1τ1+...+kMτM)?
τ∝
det[ψj(kl)]
?
a<b(ka− kb),(19)
where ψj(kl) =
spans the values l = 1,...,M and j = 0,1,...,M − 1.
Such an expression allows us to calculate all the mo-
ments and correlation functions of partial time delays by
a simple differentiation. Moreover, setting in the preced-
ing equation k1=...=kM=k and calculating the corre-
sponding limit in the right-hand side, we come to a con-
venient representation for the distribution Pu
Wigner time delay, tw=(τ1+...+τM)/MtH, for a system
with broken TRS and ideal coupling to continuum,
?∞
−∞
?∞
0dq qj−3Me−iklq−tH/q, the index l
M(tw) of the
Pu
M(tw) ∝dkeiMktwdet?ψ(n)
j
(k)?, (20)
where ψ(n)
The distribution of the Wigner time delay was earlier
calculated explicitly only for the case of M=1 [11,12,24],
when it follows from Eq. (11). Compact expression (20)
is valid for β=2 and arbitrary M [25]. For M=2, Eq. (20)
can be integrated further to yield
j
(k)≡dnψj(k)/dkn, and j,n=0,...,M−1.
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P2(tw) ∝ t−3(β+1)
w
e−β/twU?β+1
0dy ya−1(1+y)b−a−1e−zybe-
2,2β + 2,
β
tw
?,(21)
with U(a,b,z) = [1/Γ(a)]?∞
ing the confluent hypergeometrical function.
represented the above distribution (21) in a form covering
all β = 1,2,4 which will be verified below. In particular,
the asymptotic behavior at tw ≫ 1 is P2(tw) ∝ t−β−2
in agreement with the known universal tail t−βM/2−2,
which is typical for the time-delay distributions in open
chaotic systems [11,12,19,18].
To verify Eq. (21) for β=1,4, it is convenient to con-
sider a general problem of finding the distribution?
of Qs. This distribution is found to be
Here we
w
W0(?Q)
of the n×n submatrix?Q standing on the main diagonal
W0(?Q)∝θ(?Q)det(?Q)−β(M/2+n−1)−2e−βtHtr?
The particular case n=1 reproduces result (11) of the
Hamiltonian approach. Equation (22) for n=2 helps in
calculating the joint distribution ? w0(t1,t2) of two partial
U?β
?
Q−1/2. (22)
time delays t1,2=τ1,2/tHfor arbitrary M. One obtains
? w0(t1,t2)
P0(t1)P0(t2)∝
2,βM
2+β+2,
(t1t2)β/2
β
2t1+
β
2t2
?
.(23)
The knowledge of ? w0(t1,t2) allows us to find further the
prove the formula (21) for any β. As follows also from
Eq.(23), there exist nonvanishing correlations between
the partial time delays. They are, however, of differ-
ent nature as compared to the correlations between the
proper time delays (the eigenvalues of Q) which show re-
pulsion [19].
For¯S ?= 0 the matrices S and Qs cease to be statis-
tically independent variables and do correlate. There-
fore statistical properties of diagonal elements of Q in
arbitrary basis (save the eigenbasis of S) are different
from that of partial time delays, unless coupling is ideal.
Still, the JPDF w?{τc}?
nonideal coupling can be found by repeating basically
the same steps which lead to Eq. (13).
? p?{δc},{τc}?d[δ]d[τ] = ? p0
[which follows from Eq. (17)], allows us to relate w?{τc}?
and w0
?????
=d[δ]
distribution of the Wigner time delay for M=2 and thus
of the partial time delays at
The identity
?{φc},{τ(0)
c }?d[φ]d[τ(0)], to-
gether with the statistical independence of φc and τ(0)
c
?{τc}?as follows:
?
?
w({τc}) =d[δ]
∂?{φc},{τ(0)
c }?
∂?{δc},{τc}?
M
?
c=1
?????? p0
?{φc},{τ(0)
?{πρ(δc)τc}?, (24)
c }?
[πρ(δc)]p?{δc}?w0
where d[δ] means the product of differentials.
In conclusion, we suggest the transformation of the
scattering phases, allowing one to reduce the problem of
quantum chaotic scattering with statistically equivalent
channels at arbitrary coupling to that for ideal coupling.
Applications of this transformation to statistical proper-
ties of phases and those of time delays are discussed.
We are grateful to V.V. Sokolov for critical com-
ments. The financial support by SFB 237 “Unordnung
and grosse Fluktuationen” (D.V.S. and H.J.S.), RFBR
Grant No. 99–02-16726 (D.V.S.), and EPSRC Grant No.
GR/R13838/01 (Y.V.F.) is acknowledged with thanks.
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4